Applications of Derivatives Questions - JEE

If the tangent to the curve, $y = x^3 + ax - b$ at the point $(1, -5)$ is perpendicular to the line, $-x + y + 4 = 0$, then which one of the following points lie on the curve?

Let $f$ be real-valued function such that $e^{-2x}f(x) = x + 3 + \displaystyle\int_0^x \dfrac{dt}{\sqrt{t^6+1}}$ for all $x \in (-1,1)$ and let $y = g(x)$ be a function whose graph is reflection of the graph of $y = f(x)$ w.r.t. line $y = x$, then $g'(3)$ is not equal to:

Consider the following three statements for the function $f:(0,\infty)\to\mathbb{R}$ defined by $f(x)=|\log_e x|-|x-1|$: (I) $f$ is differentiable at all $x>0$. (II) $f$ is increasing in $(0,1)$. (III) $f$ is decreasing in $(1,\infty)$. Then,

Let $f$ be differentiable for all $x$. If $f(1) = -2$ and $f'(x) \geq 2$ for $x \in [1, 6]$, then

Let a, b ∈ ℝ be such that the function f given by \[f(x) = \log|x| + bx^2 + ax, \quad x \neq 0\] has extreme values at x = −1 and x = 2.Statement I: f has local maximum at x = −1 and at x = 2.Statement II: $a = -\frac{1}{2}$ and $b = -\frac{1}{4}$

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is:

Let $f:\mathbb{R}\to\mathbb{R}$ be a thrice differentiable function such that $f(0)=0,\ f(1)=1,\ f(2)=-1,\ f(3)=2$ and $f(4)=-2$. Then, the minimum number of zeros of $(3f'f''+ff''')(x)$ is _______.

The angle between the tangents to the curves $y = \sin x$ and $y = \cos x$ at a point of intersection is

We have $f(1) = -2$, $f'(x) \geq 2$ for all $x \in [1, 6]$. By LMVT, there exists $c \in (1, 6)$ such that $f'(c) = \frac{f(6) - f(1)}{5}$. Since $f'(x) \geq 2$ for all $x \in [1, 6]$, what is the minimum value of $f(6)$?

The maximum value of the function $f(x) = 2x^3 - 15x^2 + 36x - 48$ on the set $A = \{x \mid x^2 + 20 \leq 9x\}$ is:

Let a curve $y=f(x),\ x\in(0,\infty)$ pass through the points $P\!\left(1,\dfrac{3}{2}\right)$ and $Q\!\left(a,\dfrac{1}{2}\right)$. If the tangent at any point $R(b,f(b))$ to the given curve cuts the $y$-axis at the point $S(0,c)$ such that $bc=3$, then $(PQ)^2$ is equal to _____.

If the line ax + by + c = 0 is normal to the curve xy + 5 = 0, then a and b have

If f(x) − l(x) has four zeroes, where l(x) is linear and f(x) = x4 + 2x3 + cx2 + 9x + 4, then find 100b where b is the largest value of c for which the second derivative has at least two zeroes. (Hint: the second derivative f″(x) = 6x2 + 6x + c = 0 has two zeroes if and only if the discriminant 36 − 24c > 0.)

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$^2$) is equal to

The two curves $C_1: x^3 - 3xy^2 + 2 = 0$ and $C_2: 3x^2y - y^3 - 2 = 0$

The corner points of the feasible region determined by the system of linear constraints are $(0, 0)$, $(0, 40)$, $(20, 40)$, $(60, 20)$, $(60, 0)$. The objective function is $z = 4x + 3y$.Compare the quantity in Column A and Column B:Column AColumn BMaximum of $z$325

Let \(0

The function $f(x)=2x+3(x)^{2/3}$, $x\in\mathbb{R}$, has

Let $f(x) = x^2 + \dfrac{1}{x^2}$ and $g(x) = x - \dfrac{1}{x}$, $x \in \mathbb{R} - \{-1, 0, 1\}$. If $h(x) = \dfrac{f(x)}{g(x)}$, then the local minimum value of $h(x)$ is

The set of all $a\in\mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is

Let $f(x) = x^3 + x + 1$ and $g(x)$ be its inverse, then equation of tangent to $y = g(x)$ at $x = 3$ is:

$\max_{0\leq x\leq\pi}\left\{x-2\sin x\cos x+\dfrac{1}{3}\sin 3x\right\}=$

If $f(x) = \dfrac{2}{\sin x}$ and $g(x) = \dfrac{2}{\tan x}$ where \(0

The function $f(x) = \sin^4 x + \cos^4 x$ is increasing, if

Towns A and B are situated on the same side of a straight road at distances $a$ and $b$ respectively from it. Perpendiculars drawn from A and B meet the road at the points C and D respectively. The distance between C and D is $c$. A hospital is to be built at a point P on the road between C and D such that the distance APB is minimum. Find the position of P.

Let $P$ be a point on the curve $c_1: y = 2 - x^2$ and $Q$ be a point on the curve $c_2: xy = 9$, both $P$ and $Q$ in the first quadrant. If $d$ denotes the minimum distance between $P$ and $Q$, then $d^2$ is ………

Ex. 16: Statement I If differentiable function $f(x)$ satisfies the relation $f(x) + f(x+2) = 0$, $\forall x \in \mathbb{R}$, and if $\frac{1}{d}\frac{d}{dx}f(x)\bigg|_{x=a} = 6$, then $\frac{1}{d}\frac{d}{dx}f(x)\bigg|_{x=a+4000} = 6$Statement II $f(x)$ is a periodic function with period 4.

Ex. 18: If $D = 4(a^2 - 3b) = 0$, then $f(x) = x^3 + ax^2 + bx + c$

Let (2, 3) be the largest open interval in which the function f (x) = 2 log (x - 2) - x + ax + 1 is strictly e 2 increasing and (b, c) be the largest open interval, in which the function g(x) = (x - 1) (x + 2 - a) is strictly 3 2 decreasing. Then 100(a + b - c) is equal to :

The number of critical points of the function $f(x)=(x-2)^{2/3}(2x+1)$ is:

The maximum value of the function $f(x) = 2x^3 - 15x^2 + 36x - 48$ on the set $A = \{x \mid x^2 + 20 \leq 9x\}$ is:

If $2x^y+3y^x=20$, then $\dfrac{dy}{dx}$ at $(2,2)$ is equal to:

The figure shows $y = P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$. If $P''(x)$ has real roots $\alpha, \beta, \gamma$, then find $[\alpha] + [\beta] + [\gamma]$, where $[\cdot]$ denotes the greatest integer function.

If \[ f(x) = \begin{cases} -e^{-x} + k, & x \leq 0 \\ e^x + 1, & 0

The point of extremum of $f(x) = \int_0^x (t-2)^2(t-1)\,dt$ is a

If y = (1 + x)(1 + x^2)(1 + x^4) \cdots (1 + x^{2^n}), then the value of \frac{dy}{dx} at x = 0 is

Let the maximum and minimum values of $\left(\sqrt{8x-x^2-12}-4\right)^2+(x-7)^2$, $x\in\mathbb{R}$, be $M$ and $m$ respectively. Then $M^2-m^2$ is equal to _________.

If $x, y, z$ are positive real numbers, such that $x + y + z = 1$. If the minimum value of $\left(1 + \dfrac{1}{x}\right)\left(1 + \dfrac{1}{y}\right)\left(1 + \dfrac{1}{z}\right)$ is K, then find K/10.

Consider f, g and h be three real valued functions defined on ℝ.Let f(x) = sin 3x + cos x, g(x) = cos 3x + sin x and h(x) = f²(x) + g²(x)The length of a longest interval in which the function y = h(x) is increasing, is:

A rod of length 5 has ends A and B sliding along the curve $y = 2x^2$. Let $x_A$ and $x_B$ be the x-coordinates of the ends. When A is at $(0, 0)$ and B is at $(1, 2)$, find $\frac{dx_B}{dx_A}$.

If $\log_e y=3\sin^{-1}x$, then $(1-x^2)y''-xy'$ at $x=\dfrac{1}{2}$ is equal to:

The tangent at the point $(2, -2)$ to the curve, $x^2y^2 - 2x = 4(1-y)$ does not pass through the point

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice- cream layer is 1 cm , the ice-cream melts at the rate of 81 cm /min and the thickness of the ice-cream layer 3 decreases at the rate of 1 4\pi cm/min . The surface area (in cm ) of the chocolate ball (without the ice-cream layer) 2 is :

The number of points, where the curve $y=x^5-20x^3+50x+2$ crosses the $x$-axis, is _____.

Question 24: Given y = (x + √(1 + x²))ⁿ ... (i)Show that (1 + x²)(dy/dx)² = n²y²Hence find the value of n if d²x/dy² · (dx/dy)² is constant

In the above problem, the range of f(x)\ for x \in [-1, 1]\ is:

Curve is parametrically represented by $\begin{cases} x = \cos t + \ln\!\left(\tan\dfrac{t}{2}\right) \\ y = \sin t \end{cases}$ where $t$ is a parameter. The length of the tangent drawn to the curve at the point where its $x$-coordinates is equal to its $y$-coordinates is:

Curve is parametrically represented by $\begin{cases}x=\cos t+\ln\!\left(\tan\dfrac{t}{2}\right)\\y=\sin t\end{cases}$ where $t$ is a parameter. The length of the tangent drawn to the curve at the point where its $x$-coordinates is equal to its $y$-coordinates is:

The tangent to $y = f(x)$ at $x = 0$ has slope equal to:

Consider the triangles with vertices $A(2,1)$, $B(0,0)$ and $C(t,4)$, $t\in[0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha+21\beta$ is equal to ___________.

Applications of Derivatives Questions (514)